## Saturday, November 20, 2010

### Circular Motion-2

Vertical Circular Motion Problems

1. A ball with a mass of 130 g is swung at the end of a string 93.0 cm in length. The ball is whirled in a vertical circle at 4.00 revolutions per second.
a. What is the tension on the string at the bottom of the loop?
b. What is the tension on the string at the top of the loop?

2. A jet fighter pilot knows he is able to withstand an acceleration of 9g before blacking out. The pilot points his plane vertically down while traveling at Mach 3 speed and intends to pull up in a circular maneuver before crashing to the ground.

a) Where does the maximum acceleration occur in the maneuver?

b) What is the minimum radius the pilot can take?

3. A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to 3.0 times her weight as she goes through the dip. If r = 25.0 m, how fast is the roller coaster traveling at the bottom of the dip?

4. What is the apparent weight of a 75-kg person driving a car with a constant speed of 12 m/s over a bump with a circular cross-section and radius of curvature of of 35 m?

5. What is the minimum speed of a roller coaster at the top of a 39.0 m vertical loop if the passengers are "weightless" at that point.

6. A ball of a mass 4.0 kg is attached to the end of a 1.2 cm long string and whirled around in a circle that describes a vertical plane.

a. What is the minimum speed that the ball can be moving at and still maintain a circular path? Provide a free body diagram.

b. At this speed, what is the maximum tension in the string? Provide a free body diagram.

c. If the ball was swung in a horizontal circle at this speed, what angle would the string make with the vertical?

7. How do you find the tension in the string of a ball traveling in a vertical circle at the 45 degree angle?

8. A hill is in the shape of an arch having the radius of curvature of 41. m. What is the maximum speed that a car can travel across the hill without 'getting some air'?

Banked Curves

1. Determine the minimum angle at which a road should be banked so that a car traveling at 20.0 m/s can safely negotiate the curve if the radius of the curve is 200.0 m.

2. If a curve with a radius of 65 m is properly banked for a car traveling 75 km/h, what must be the coefficient of static friction for a car not to skid when traveling at 90 km/h?

3. A Car is driven around a circle with a radius of 200m, bank angle 10 degrees. The static frictional coefficient is 0.60. Calculate the maximum velocity the car can travel.

4. An airplane is flying in a horizontal circle at a speed of 460 km/h. If its wings are tilted 40° to the horizontal, and force is provided by lift that is perpendicular to the wing surface. What is the radius of the circle?

### Circular Motion-1

A. Speed and Acceleration

1. A 2.0 kg mass swinging at the end of a 0.50 m string is traveling 3.0 m/s. What is the
a. centripetal acceleration of the mass?
b. centripetal force on the mass?

2. A person standing at the Earth's equator has what rotational speed? (R = 6.38 x 106 m)

3. A building is located 28º north of Earth's Equator. What distance does it travel as a result of the Earths rotation?

4. A planet has a radius of 6.04*106 m and free fall acceleration of 9.9 m/s2. What would be the tangential speed of a person standing at the equator, if its rotation increased to the point that the centripetal acceleration was equal to the gravitational acceleration?

B. Horizontal Circular Motion Problems

1. A stopper tied to the end of a string is swung in a horizontal circle. If the mass of the stopper is 13.0 g, and the string is 93.0 cm, and the stopper revolves at a constant speed 10 times in 11.8 s,
a. what is the tension on the string?
b. what would happen to the tension on the string if the mass was doubled and all other quantities stayed the same?
c. what would happen to the tension on the string if the period was doubled and all other quantities stayed the same?

2. A rock is whirled on the end of a string in a horizontal circle of radius R and period T. If the radius is halved while keeping the period constant, what happens to the centripetal acceleration of the rock?

3. A boy is whirling a yo yo on a string in a horizontal circle. What happens to the tension on the string when he whirls it twice as fast?

4. A 0.19 m cord passes through a hole in a table.. The cord attaches a mass m = 2.8 kg on the frictionless surface to a hanging mass M = 7.9 kg. Find the speed with which m must move in a circle in order for M to stay at rest?

5. A clock rests horizontally on a table with a pebble balanced at the end of its 1.0 cm long second hand.. What is the minimum coefficient of static friction which would allow the pebble to stay there without slipping?

6. What is the smallest radius of an unbanked (flat) track around which a motorcyclist can travel if her speed is 25 km/h and the coefficient of static friction between the tires and the road is 0.28?

7. A rock of mass 4.0*102 g is tied to one end of a string that is 2.0 m in length and swung around in a circle whose plane is parallel to the ground.

a. If the string can withstand a maximum tension of 4.5 N before breaking, what angle to the vertical does the string reach just before breaking?

b. At what speed is the rock traveling just as the string breaks?

8. A 1050 kg car travels around a turn of radius 70 m on a flat road. If the coefficient of friction between tires and road is 0.80 what is the maximum speed the car can travel without slipping? Does this result dependent on the mass of the car?

9. Two ice skaters of equal mass grab hands and spin in a circle once every four seconds. Their arms are 0.76 m long, and they each have a mass of 55.0 kg. How hard are they pulling on one another?

10. A hollow vertical cylinder with radius R spins about its vertical axis of symmetry. A stone is held to the inner cylinder wall by static friction. Express the period of rotation in terms of the radius and the coefficient of friction, μs.

11. A coin slides around a horizontal circle at height y inside a frictionless hemisphere bowl of radius R. Derive the coin’s velocity in terms of R, y and g.

12. A train travels at a constant speed around a curve of radius 225 m. A ceiling lamp at the end of a light cord swings out to an angle of 20.0º throughout the turn. What is the speed of the train?

### Mutual Inductance and Transformers

Transformers make use of mutual inductance. In this process the changing magnetic field produced by the "primary" coil induces electric potential in the "secondary" coil. The primary coil is the one attached to an alternating current power source. An alternating current ("AC") increases then decreases in one direction then reverses, increasing then decreasing in that direction. The changing current creates a changing magnetic field. The changing magnetic field induces electric potential in the nearby secondary coil, which is attached to a load.
Transformers consist of stationary solid core coils near one another.
Every one who lives in a developed country depends on transformers. Transformers "step down" high voltage power delivered from hydro dams or other far away generators to 240 V delivered to the house. Other devices such as televisions and computer monitors use transformers to step up voltage from the 120 V wall outlets supply to the 2500 V needed by CRT's (television tubes).

Whether the voltage is stepped up or down and by how much depends on the relative number of turns on the two coils. The basis of this wizardry is the Law of Conservation of Energy. Power in an electric circuit is directly proportional to current and voltage:
P = IV .
Ideally the power developed in the secondary coil would be the same as developed in the primary:
Pp = Ps .

Ignoring the heat loss that occurs we will take this as an accurate statement: therefore,
IpVp = IsVs.

Rearranging this equality we are able to compare the relative voltages and currents in the primary and secondary coils:
Ip/Is = Vs/Vp .

What this last equality is saying is that you don't get something for nothing. If you use a transformer to double the voltage, you only get half the current. At best, ignoring heat losses, you get the same power from the secondary as delivered to the primary.

As indicated by Faraday's law, the electric potential induced is directly proportional to the number of loops or

turns in the coil. Therefore,
Ns/Np = Vs/Vp = Ip/Is .

In functioning transformers, hundreds or thousands of turns are used. The transformer depicted above has

Np = 4 and Ns = 2. Therefore

Ns/Np = Vs/Vp = 0.5

Therefore, if 120 V was delivered to the primary coil,
0.5 = Vs/(120 V)

the voltage delivered to the load in the secondary coil would be
Vs = 60 V. This would be a step-down transformer.
If the source delivered 2 A of current to the primary coil (Ip = 2 A) then

Ns/Np = Ip/Is and

0.5 = (2 A)/Is and

Is = 4 A .

(At best we get half the voltage; twice the current. Nothing gained; nothing lost.)

### Transformers

1. A transformer is used to change a 120 V, 3 A current to 2500 V.
a. What kind of transformer is this?
b. What is the ratio of secondary turns to primary turns?
c. What current would be developed in the secondary coil?

2. A step-up transformer has 100 turns on the primary coil and 500 turns on the secondary coil. If this transformer is to produce an output of 4300 V with a 12 mA current, what input current and voltage are needed?

3. The average emf induced in the secondary coil is 0.12 when the current in the primary coil changes from 3.4 to 1.6 A in 0.14 s. What is the mutual inductance of the coils?

4. A circular coil with 233 turns and a diameter of 23.5 cm rotates about a vertical axis at 1250 rpm. The coil is situated in a magnetic field having a horizontal component of 3.80x10-5 T, and a vertical component of 2.85 x10-5 T. What is the maximum EMF produced in the coil?

5. The current in an air-core solenoid is reduced from 3.99 A to zero over 5.9s. The solenoid has 2000 turns per meter and a cross-sectional area of 0.131 m2. Surrounding the solenoid near the center of its length is a second coil of 50 turns.

a. What is the magnitude of the induced emf in the second coil?

b. If the resistance of the second coil is 0.00409 ohm what is the induced current?

6. A 200-turn air-core solenoid with a cross-sectional area of 100 cm2 has a resistance of 5.0 ohms. The ends of the wire are joined together to close the circuit. A 1.1 T magnetic field is directed through the coil perpendicular to its cross-sectional area. Over a period of 0.1s, the field is reversed. What average current flows through the coil during that period?

### Electric Fields

1. Starting with a neutral electroscope, show the charge distribution and action of the leaves when the electroscope is first touched by a positively charged object, and then by a large neutral object.
2. Starting with a neutral electroscope, show the charge distribution and action of the leaves when the electroscope is brought near a negatively charged object.
3.
a. Two objects are identical in every way except that one is neutral, and the other has 2 excess electrons. Show what happens to the distribution of charges when the two objects are brought into contact and then released.
b. If the charge on the electron is "-1" what is the charge on each of the two objects after they are separated?
4.
a. Two objects are identical in every way except that one is deficient by two electrons, and the other has 4 excess electrons. Show what happens to the distribution of charges when the two objects are brought into contact and then released.
b. If the charge on the electron is "-1" what is the charge on each of the two objects after they are separated?
5. Two neutral identical objects , A and B, are in contact and brought near a negatively charged object, C. While in the presence of C, A and B are separated. What are the relative charges of A and B? Draw diagrams to show the charge distributions at each step.

6. A charge of -2 x 10-6 C experiences a force of 0.08 N [left]. What is the electric field at that point?
7. A charge of +3.0 x 10-6 C is 0.25 m away from a charge of -6.0 x 10-6 C.
a. What is the force on the 3.0 x 10-6 C charge?
b. What is the force on the -6.0 x 10-6 C charge?
8. Three charges, q1 = 4 x 10-6 C, q2 = -2 x 10-6 C, and q3 = 5 x 10-6 C are placed at the corners of a square with

sides 0.30 m. What is the field at at the fourth corner?

9. A charged droplet of mass 5.88 x 10-10 kg is hovering motionless between two parallel plates. The parallel plates have a potential difference of 24000 V and are 2.00 mm apart. What is the charge on the particle? By how many electrons is the particle deficient?

10. Four point charges form the vertices of a square with sides = L. Two diagonally opposite charges have a charge of 2.25 C each. The other two charges are identical to each other and each have a charge, q. If there is no net force on either of the 2.25 C points, what is the value of q?

11. Two point charges lie on the x-axis. A charge of 9.9 C is at the origin, and a charge of -5.1 C is at x=10cm.

a. At what position x would a third charge q3 be in equilibrium?
b. Does your answer to part a depend on whether q3 is positive or negative? Explain.

12. Two particles each with a positive charge of q are placed on the vertices of a square having sides a. A third particle with a positive charge Q is placed at the center of the square. What is the force on the particle at the center of the square?

13. A charge of 6.00*10-9 C and a charge of -3.00*10-9 C are separated by a distance of 60.0 cm. Find the position at which a third charge of 12.0*10-9 C can be placed so that the net electrostatic force on it is zero.

14. An electron enters a region where the field strength is 3.0*106 N/C. (a) What is the electron's acceleration? (b) Starting from rest, how far does the electron travel to acquire 10% of the speed of light?

15. Four point charges, each of magnitude 2.34*10¹ C, form a square with sides 40.8 cm. If three of the charges are positive and one is negative, find the magnitude of the force experienced by the negative charge.

16. Two 24-g spheres are each attached to the bottom of very light 78 cm wires. When the wires are joined at the top, they each form an angle of 30 degrees to the vertical. What is the total charge on the spheres?

17. Two point charges have a total charge of 560 μC. When placed 1.10m apart, the force each exerts on the other is 22.8N and is repulsive. What is the charge on each?

18. Explain how to calculate the amount of free charge in a wire.

19. A sphere with a charge of -50 is centered within a hollow sphere having a charge of -100. Describe the distribution of charges.

20. A square with sides 52.5 cm is formed by a +45.0 x 10-6 C charge at one corner and -27.0 x 10-6 C charges at each of the other corners. What is the electric field at the center?

21. A -3.5 x 10-10 C point charge is fixed near the Earth's surface. An electron is placed near the point charge so that the electric force acting on the electron cancels the electron’s weight. Where is the electron relative to the point charge?

22. Three point charges, each +4.6 μC, form a straight line. Charge A is 1.8 m from the central charge, B. Charge C is 2.2 m from charge B. What is the magnitude and direction of the net force on each charge?

23. An 4.50 μC electric charge is in an electric field with a y-component Ey = 4000 N/C, an x-component Ex = 700 N/C and a z-component Ez = 0. What are the magnitude and direction of force on the charge?

24.Two charges, +q and 4q, are 1 m apart. What are the location, magnitude and sign of a third charge, Q, placed so that the entire system is at equilibrium?

25. Find the electric field midway between charges of +.000000030 C and +.000000060 C 30.0 cm apart.

26. A -2.00 μC forms the apex of a triangle, while two +5.00 μC charges form the base. One of the +5.00 μC charges is 20.00 cm from the apex, and the third charge is 8.75 cm away from the apex. The angle at the apex is 1.396 rad. Find the net force on the third charge.

27. Two parallel plates 2.1mm apart have a 36V potential difference. (i) What is the electric field strength between the plates? (ii) Sketch the electric flux lines between the plates, and show the direction of the field. (iii) Suggest three ways to increase capacitance. (iv) Find the force on a +180nC particle placed midway between the plates, and the energy required to move the particle 0.7mm towards the positively charged plate.

28. Explain how to calculate the magnitude and direction of the acceleration of a particle given the electrical field intensity.

29. A +2.0 μC charge is located on the x-axis at +0.3 m and another at -0.3 m. A third charge, +4.0 μC, is located on the y-axis at +0.4m. Find

a. the net force on the third charge

b. the electric field at (0,-0.4m)

c. the potential at (0,-0.4m)

30. Suppose that equal and opposite charges were placed on the Earth and the Moon. What amount of charge on each would supply an electrical force equal to the gravitational force between them?

31. Point charges, -1q,-2q,-3q......,-12q, are fixed at the corresponding positions on the face of a clock. What is the direction of the electric field?

32.

An electric dipole is dagrammed as formed from two electrostatic charges along the x axis equidistant from the origin.

Two non-coincident point charges on the x axis are each separated by a distance, a, from the origin. Show that at a distant point along the x axis the electric field is given by

Ex = 4keqa/x

33. Find the total electric flux through a spherical shell placed in a uniform electric field.

34. A tiny plastic sphere (mass = m, charge = –q) hovers above a large horizontal plastic sheet having a uniform charge density on its surface. Use Gauss' Theorem to find the sheet’s charge per unit area.

35. An electron with an initial kinetic energy of 1.60 × 10–17 J decelerates to rest over a distance of 10.0 cm. What are the magnitude and direction of the electric field that stopped the electron?

### Electromagnetic Induction

1. An electron enters a 4.0 T field with a velocity of 5.0 x 105 m/s perpendicular to the field. What is the radius of curvature of its path?

2. A conducting rod of 25 cm is pushed across a magnetic field along a U-shaped wire at a constant speed of 2.0 m/s. The field is directed away from the observer and is 4.00 T. A current of 8.00 A is induced in the circuit.
A conducting rod of 25 cm is pushed across a magnetic field along a U-shaped wire at a constant speed of 2.0 m/s.
a. What is the potential difference induced in the circuit?
b. What is the resistance of the circuit?
c. What is the force used to push the rod?
d. In what direction is current flowing in the rod?

3. A wire with a linear density of 1 g/cm moves horizontally to the north on a horizontal surface with a coefficient of friction 0.2. What are the magnitude and the direction of the smallest magnetic field that enables the wire to continue in this fashion?

4. An electron is accelerated from rest through a potential difference of 18 kV, and then passes through a 0.34-T magnetic field. Calculate the magnitude of the maximum magnetic force acting on the electron.

5. A 0.0017 T magnetic field and a 5.7 x 103 N/C electric field both point in the same direction. A positive 2.0-mC charge moves at a speed of 2.9 x 106 m/s perpendicular to both fields. Determine the magnitude of the net force on the charge.

6. What is the magnitude of the magnetic force on an electron moving 5.0 x 104 m/s perpendicular to a uniform magnetic field of .20T?

7. What speed would a proton need to orbit 1000 km above the Earth along the magnetic equator where the magnetic field intensity is 4.00x10-8 T?

8.

a. Find the magnetic flux density 3.1 mm away from a long straight wire carrying a 1.2 A current.

b. What force per meter would act on a long straight wire 3.1mm away from and parallel to the wire in part (a) and carrying a 4.5A current?
c. What force would act on a 37 μC charged particle 3.1mm from the wire in part (a) and moving away from the wire at 6.5 m/s?

9. Find the magnetic flux density in the center of a 4.0 cm long air-core solenoid made with 4900 turns of wire and carrying a 2.5A current.

10. Express the attenuation distance for a plane electromagnetic wave in a good conductor in terms of the conductivity σ, permeability μ0, and frequency ω.

### Magnetic Fields

1. An alpha particle (two protons and two neutrons) traveling east at 2.0 x 105 m/s enters a magnetic field of 0.20 T pointing straight up. What is the force acting on the alpha particle?

With the fingers of the right hand pointing straight up, and the thumb pointing east, the palm points south.
F = qvBsinø = (2 x 1.6 x 10-19 C)(2.0 x 105 m/s)(0.20 T)sin90º = 1.28 x 10-14 N [S].

2. An electron traveling to the left, moves into a magnetic field directed toward the observer. Trace the path of the particle, assuming it eventually leaves the field.
An electron traveling to the left, moves into a magnetic field directed toward the observer.

The moment the electron enters the field, it experiences a force perpendicular to its velocity. The electron follows a circular path until it leaves the field.
The moment the electron enters the field, it experiences a force perpendicular to its velocity.

3. A horizontal conductor is carrying 5.0 A of current to the east. A magnetic field of 0.20 T pointing straight up cuts across 1.5 m of the conductor. What is the force acting on the conductor?

With the fingers of the right hand pointing straight up, and the thumb pointing east, the palm points south.
F = BILsinø = (0.20 T)(5.0 A)(1.5 m)sin90º = 1.5 N [S].

4. A 50.0 cm horizontal section of conductor with a mass of 8.00 g is in a 0.400 T magnetic field directed to the west. What are the magnitude and direction of current required to make this section of the conductor seem weightless?

The magnetic force must be opposite and equal to the weight of the section of the conductor.
With the fingers of the right hand pointing west, and the palm facing straight up, the thumb points north.
The weight of the conductor is mg = (0.00800 kg)(9.8 N/kg) = 0.0784 N .
The magnetic force on the conductor is
F = BILsinø , so
0.0784 N = (0.400 T)I(0.50 m)sin90º
I = 0.392 A [N]

### Work and Potential Energy

1. How much work is done by a crane lifting a 200.0 kg crate from the ground to a floor 21.0 m above the ground. What is the change in gravitational potential energy of the crate?
2. A 25-kg box slides, from rest, down a 9.0-m-long incline that makes an angle of 15° with the horizontal. The speed of the box when it reaches the bottom of the incline is 2.4 m/s.
a. What is the coefficient of kinetic friction between the box and the surface of the incline?
b. How much work is done on the box by the force of friction and
c. What is the change in the potential energy of the box?
3. A 40.0-kg wagon is towed up a hill inclined at 18.5º with respect to the horizontal. the tow rope is parallel to the incline and has a tension of 140N in it. Assume that the wagon starts from rest at the bottom of the hill, and neglect friction. How fast is the wagon going after moving 80 m up the hill?
4. A 25.6kg child pulls a 4.81kg toboggan up a hill inclined at 25.7° to the horizontal. The vertical height of the hill is 27.3 m. Friction is negligible. Determine how much work the child must do on the toboggan to pull it at a constant velocity up the hill.
5. An 81.0-kg in-line skater does +3500 J of nonconservative work by pushing against the ground with his skates. In addition, friction does -710 J of nonconservative work on the skater. The skater's initial and final speeds are 2.50 m/s and 1.60 m/s, respectively. (a) Has the skater gone uphill, downhill, or remained at the same level? (b) Calculate the change in height of the skater.
6. A solid object with mass (m) is initially at rest. An applied constant vertical force (F) causes the object to reach an upward speed (V), and total displacement (h). Use newton's second law to derive an expression (in terms of m,g,h, & V) for the work.

1. A cart moving along a track 1.00 m above the floor at 3 m/s eventually reaches a higher plateau What is the maximum height of the plateau above the floor?

2. A 10.5 g bullet strikes a pendulum that consists of a block of wood of mass 3.00 kg suspended by a cord. The bullet gets embedded in the block. How fast was the bullet traveling just before impact to raise the block by 0.220 m?

3. Sheila, running 5.3m/s, grabs a vine hanging vertically from a tall tree.
a. How high can she swing upward?
b. Does the length of vine affect the answer?

4. A roller coaster at the top of a 39.0 m high vertical loop is traveling 13.8 m/s. Find the maximum speed of the cars as they move through the bottom of the loop.

5. Analyze the motion of a simple swinging pendulum in terms of energy, (a) ignoring friction; and (b) taking friction into account. Explain why a grandfather clock has to be wound up.

6. A ball is attached to a horizontal cord of length L whose other end is fixed.
a. If the ball is released, what will be its speed at the lowest point of its path?
b. A peg is located a distance h directly below the point of attachment of the cord. If h= 0.080L, what will be the speed of the ball when it reaches the top of the circular path about the peg?

7. A projectile is fired at an upward angle of 45.0 degree from the top of a 265 m cliff with a speed of 185 m/s. What will be its maximum speed of impact with the ground below?

8. If a projectile is launched from Earth with a speed equal to the escape speed, how high above the Earth's surface is it when its speed is one third the escape speed?

9. Two pieces of space debris, each with a mass of 0.116 kg, are separated by a distance of 380 m. If re released from rest, what speed do they have when their separation has decreased to 171 m? Ignore the gravitational effects from any other objects.

10. A baseball is thrown first with an initial upward velocity of + 4.0 m/s. Later, it is thrown from the same height but with and initial downward velocity of -3.0m/s. How do the impact velocities of the baseball with the ground differ? What is its acceleration in each case?

11. A 200 g ball is thrown upwards with an initial kinetic energy of 10 Joules. What maximum height will the ball attain?

12. Bruce grasps the end of a 20.0 m long rope attached to a tree and swings. If the rope starts at an angle 35 degrees with the vertical, what is Bruce's speed at the bottom of the swing?

13. Jack and Jill, whose total mass is 120 kg, sit on a swing at the end of a 5m long rope. Initially the rope attached to their swing makes an angle of 36 degrees with the horizontal. At the bottom of the arc, Jill, whose mass is 52 kg, steps off. What is the maximum height Jack can reach as the swing continues?

14.
a. A 1.9-kg block slides down a curved, frictionless ramp. The top of the ramp is 1.5 m above ground; the bottom of the ramp is 0.25 m above the ground. The block leaves the bottom of the ramp moving horizontally. What horizontal distance away from the base of the ramp does it land?
b. Suppose friction on the ramp does -9.7 J of work on the block. What horizontal distance away from the base of the ramp does it land?

15. A ball bounces upward from the ground with a speed of 16 m/s and hits a wall with a speed of 12 m/s How high above the ground does the ball hit the wall? Ignore air resistance.

16. From what height would a car have to be dropped to have the same kinetic energy that it has when being driven at 100 km/h?

17. A 135 m long ramp is to be built for a ski jump. If a skier starting from rest at the top is to have a speed no faster than 19m/s at the bottom, what should be the maximum angle of inclination?

18. Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.

19. Two objects, m1 = 4.50kg and m2 = 3.00kg, are connected by a light string passing over a light frictionless pulley. The object of the mass 4.50kg is released from rest 4.50m above the ground. Using the principle of conservation of energy, determine the speed of the 3.00kg object just as the 4.50kg object hits the ground.

20.
(a) What is the escape speed on a spherical asteroid whose radius is 525km and whose gravitational acceleration at the surface is 2.7m/s2?
(b.)How far from the surface will a particle go if it leaves the asteroid's surface with a vertical speed of 1000m/s?

21. In a looping the loop setup, an object of mass m is released from rest from A with the initial height h and loops the loop in the circular track of radius R.
a) Write an expression for the initial mechanical energy at A in terms of m, g, h
b) Write an expression for energy at point B at the top of the vertical circle in terms of mass m, velocity v, radius R and g
c) If in the absence of friction the object just manages to loop the loop without losing contact with the track, what is the minimum height h from which you will need to release the object? Write the expression in terms of R .

22. A 75 kg parcel falls out of a window to a sidewalk 1 m below.
a. With what speed does it impact the pavement?
b. If the packaging provides 0.50 cm of cushioning, calculate the average force exerted on the parcel by the ground in this situation.

23. Roy was transporting balls in the trunk of a car to a clubhouse. Two boxes on the floor of the trunk each contained an equal number of balls. The balls were identical except that all the balls in one box were dimpled, while all the balls in the other box were smooth. Upon arriving, Roy realized there were no lids on the boxes, and found balls all over the trunk of the car. He observed that more dimpled balls escaped than smooth balls. Why would more dimpled balls escape than smooth balls? There was nothing else in the trunk.

### Work and Kinetic Energy

1. A boy pushes a 5.00 kg cart in a circle, starting at 0.500 m/s and accelerating to 3.00 m/s. How much work was done on the cart?
2. A 30.0 kg box initially sliding at 5.00 m/s on a rough surface is brought to rest by 20.0 N of friction. What distance does the box slide?
3. A 1000.0 kg truck accelerates from 20.0 m/s to 25.0 m/s over a distance of 300.0 m. What is the average net force on the truck?
4. A space ship of mass 5.00 ×104 kg is traveling at a speed 1.15 × 104 m/s in outer space. Except for the force generated by its own engine, no other force acts on the ship. As the engine exerts a constant force of 4.00 × 105 N, the ship moves a distance of 2.50 × 106 m in the direction of the force of the engine.
a. Determine the final speed of the ship using the work-energy theorem.
b. Determine the final speed of the ship using the kinematics equations.

5. A force of 6.0 N is used to accelerate a mass of 1.0 kg from rest for a distance of 12m. The force is applied along the direction of travel. The coefficient of kinetic friction is 0.30. What is the
a. work done by the applied force?
b. work done by friction?
c. kinetic energy at the 12-m mark?

6. A 0.600-kg particle has speed of 2.00 m/s at point A and kinetic energy of 7.50 J at point B. What is
a. its kinetic energy at A?
b. its speed at B?
c. the total work done on the particle as it moves from A to B?

7. Two carts, one twice the mass of the other, experience the same force for the same time. What is their difference in momentum? What is their difference in kinetic energy?

8. A weapon fired a 25.8-kg shell with a muzzle speed of 880 m/s. What average force acted on the shell?

9. A catcher stops a 91 mi/h pitch in his glove, bringing it to rest in 0.00179 m. If force exerted by the catcher is 785 N, what is the mass of the ball?

10. A 7.80 kg package was dropped onto a flatbed moving horizontally at 1.60 m/s. The coefficients of friction are as follows: µs= 0.470 and µk = 0.150 . How far does the package slide on the flatbed?
11. A 12.4 g bullet is fired horizontally into a 96 g wooden block initially at rest on a horizontal surface. After impact, the block slides 7.5 m before coming to rest. If the coefficient of kinetic friction between block and surface is 0.650, what was the speed of the bullet immediately before impact?

12. A toy cart moves with a kinetic energy of 30 J. If its speed is doubled, what is its kinetic energy?
13. A car traveling 59 miles/hour locks its brakes until the car reaches 42 miles/hour. The mass of the car is 79.2 slugs. Calculate the energy lost to friction.
14. A 70 kg diver steps off a 10 m tower and drops from rest straight down into the water. If he comes to rest 5 m beneath the surface, determine the average force exerted by the water.
15. A 1300-kg car slows from 18 m/s to 15 m/s through a distance of 30 m. (a) Was the net work done on the car positive, negative, or zero? Explain. (b) Find the magnitude of the average net force on the car in the sandy section.

### Work done, Energy and Power

1. A man pushes against a car stuck in a snow bank while his date sits nervously behind the steering wheel trying not to make the tires spin. [Other distracting details are usually given, often including numeric information regarding force, weight and so on.] However, the car does not move. How much work did he do on the car?
Answer: The man did no work on the car since d=0. He may have burned calories, converting chemical energy into heat, but still, the car did not move.
2. Sally carries a text book under her arm along a horizontal path. [Distracting information is often also given such as the weight or mass of the text book, the distance or more perversely the path taken implying some distance must be calculated and so on.] How much work was done on the text book?
Answer: None, since both gravity and the force Sally exerted against gravity are perpendicular to the distance the book moved. (cosø = 0 so W = 0).
3. An asteroid traveling at constant velocity out of reach of gravitational fields [etc.]... How much work is done on the satellite?
Answer: None, since F = 0, W = 0.

Another type of question gives the wrong angle.
4. A 200.0 kg load on frictionless coasters is pushed 5 m along a ramp that makes an angle of 20º to the ground. How much work was done on the cart?
Answer: Drawing a diagram is a must for any problem involving 2 dimensions or 2 or more forces.

When you do this, the glitch becomes obvious: the angle is not between the force and the distance. The angle between the force and the distance is 70º.
The force applied against gravity is F = mg = (200.0 kg)(9.8 N/kg) = 1960 N
Therefore, W = Fdcosø = 1960(5)(cos70º) = 3352 J
5. A 70 kg cart is pushed for 50 m with a constant velocity upon a 45º frictionless incline. Find the work done on the cart.
6. A 100 kg object is pulled vertically upward 5.0 m by a rope with an acceleration of 1.0 m/s2. Find the work done by the tension force in the rope.

7. A 0.40 kg ball is thrown vertically upward with a speed of 30 m/s. The ball reaches a height of 40 m. What is the energy dissipated due to air friction?

8. If 4 kW of power is dissipated for 30 min, how much energy was involved?

9. A tug pulls on a barge with a constant force of 2.50*106 N, 19° west of north. Another tug pulls on the same barge with the same force but 19° east of north. . What is the total work they done on the barge it is pulled 0.74 km toward the north?

10. A conveyor belt moves for 2.0 m horizontally, then for 2.0 m down an incline angled 10° from the horizontal. A 2.0 kg box is moved by the belt at 0.50 m/s without slipping. At what rate is work being done on the box
a. as the box moves up the 10° incline
b. as the box moves horizontally

11. A 40 kg case is pushed across a floor at a steady speed of 1.5 m/s. When the pushing stops, the case slides a further distance of 1.2 m before coming to rest. Calculate:
a. The frictional force acting on the case when it slides.
b. The work done per second to push the case at a steady speed of 1.5 m/s.

12. A generator harnesses water flowing through a tube 12.5 feet in diameter. About 50% of the kinetic energy of the water is converted to electricity. How fast must the water flow to produce 52 MW of electrical power?

13. A 130 N suitcase is dragged a distance of 250m at a constant speed. A force of 60 N is exerted at an angle 40 degrees above the horizontal.
a. How much work is being done?
b. How much work is done by friction?

14. How many kilocalories of food energy does a person with a metabolic rate of 97.0 W burn per day?

15. A 1000 kg rocket generating 80 kW of power gains altitude at 3.0 m/s. What percentage of the engine power is being used to make the rocket climb?

16. How does a person burn calories by just breathing?

17. Calculate the work done by a force acting through a distance, d, if the force changes over the distance according to

F = kx4.

18. Can the normal force on an object ever do work?

19. If 6 million J of energy is consumed in a day, what is this rate of energy consumption in watts?

20. A force of 109 N is applied along along a lawnmower's handle, which is 14.7 degrees above the horizontal. If 61.9 W of power was developed for 39 s, what distance is the roller pushed?

21. Suggest a method to mesure the power in your legs. Outline the calculations involved.

22. A 1900 kg car experiences a combined force of air resistance and friction that has the same magnitude whether the car goes up or down a hill at 27 m/s. Going up a hill, the car's engine needs to produce 47 hp more power to sustain the constant velocity than it does going down the same hill. At what angle is the hill inclined above the horizontal?

23. Find the work done by pushing a mass of 700 kg through a distance of 4.5m along a surface if the coefficient of friction is 0.2. Assume the force is applied horizontally.

24. At a speed of 70 km/h, the average frictional force on a light car is 1050 N. What power must the motor generate to maintain this speed?

25. A winch is used to slide a 350 kg load at constant speed 3.50 m down a 27.0° incline. The rope between the winch and the load is parallel to the incline. The coefficient of kinetic friction between the ramp and the load is 0.40. Calculate the force exerted by the winch, the work done by the winch on the load, the work done by the friction force, the work done by the force of gravity, and the net work done on the load.

26. A 60 kg person developed 70 W of power during a race, dissipating about 0.60 J of mechanical energy per step per kilogram of body mass. If each stride was about 1.5 m, how fast was the person running?

### Free fall

A. Drop /Throw Down Problems

1. A peso is dropped into a well and it falls for 3 seconds before hitting the water. What is the peso's average speed during its 3 second drop?

2. A rock is dropped from the top of an overhang and strikes the ground 6.5 seconds later. How high is the overhang in meters?

3. It takes 0.210s for a dropped wrench to travel past a poster that is 1.35 meters tall. How high above the top of the poster was the wrench released?

4. A falling paratrooper (100 kg with parachute) experiences air resistance equal to 25% of his weight. What is his acceleration?

5. Consider a transparent elevator accelerating upward with acceleration equal to that of the gravity. If a rock was dropped inside the elevator, what would an observer on the ground see the rock do?

B. Throw Up Problems

1. You throw a ball downward from a window at a speed of 2.0 m/s. The ball accelerates at 9.8 m/s2.
a. How fast is it moving when it hits the sidewalk 2.5 m below?
b. If you throw the same ball up instead of down, how fast is it moving when it hits the sidewalk?

2. A ball is thrown straight up with a speed of 4.6 m/s. How long does the ball take to reach its maximum height?

3. A round is launched straight up at 460 m/s. How long will it take it to reach its apex and how high will that be? (air resistance may be neglected.)

4. What height will a dart achieve 7 seconds after being blown straight up at 50 m/s?

5. An apple thrown straight upward rises to 24 m above its launch point. At what height has apple's speed decreased to one-half of its initial value?

6. A stone is flung straight up from a point 1.50 m above the ground and with an initial speed of 19.6 m/s.
a. What is the stone's maximum height above the ground?
b. How much time passes before the stone hits the ground?

7. A blimp is hovering above the ground. When the pilot drops a sandbag overboard, the blimp rises with a constant velocity of 2 m/s. At the moment the sandbag hits the ground, the blimp is 50 m above the ground.
a. How far above the ground was the blimp when the sandbag was dropped?
b. When the sandbag is halfway to the ground, what is its acceleration?

8. A helicopter is ascending vertically with a speed of 5.00 m/s. At a height of 105 m above the ground, a package is dropped from a window. How much time does it take for the package to reach the ground?

9. A rock is thrown up at an initial velocity of 9.8 m/s. What is the time it takes to hit the ground?

C. Catch Up Problems

1. An archer shoots an arrow with an initial velocity of 21 m/s straight up from his bow. He quickly reloads and shoots another arrow in the same way 3.0 s later. At what time and height do the arrows meet?

2. A hoist is lifting a naturalist to the top of a cliff at 2.03 m/s vertically. The naturalist suddenly realizes she has left her pet rock behind. A friend picks it up and tosses it straight upward. If the naturalist is 2.50 m above her friend, what is the minimum initial speed the pet rock must have to reach the naturalist?

3. A boy shoots a rock from his slingshot at a target just as the target drops from a tree branch. Should the boy aim a little above the target, a little below the target, or straight at the target?

### Projectiles

1. A golf ball is projected with a horizontal velocity of 30 m/s and takes 4.0 seconds to reach the ground. (Assume g= 10 m/s² and the air resistance is negligible.) Calculate: the height from which the golf ball was projected. The magnitude of the golf balls' vertical velocity component just before hitting the ground. The horizontal velocity component. Resultant velocity just before the object strikes the ground. The horizontal component of the object's displacement.

2. Erica kicks a soccer ball 12 m/s at an angle of 40 degrees above the horizontal.

a. What is the ball's maximum height?
b. What is the ball's maximum range?
c. With what velocity does the ball strike the ground?
d. What are the ball's acceleration and velocity at the top of its rise?

A Horizontal Projectile Motion

1. Erica kicks a soccer ball 12 m/s at horizontally from the edge of the roof of a building which is 30.0 m high.
a. When does it strike the ground?
b. With what velocity does the ball strike the ground?

2. A car drives straight off the edge of a cliff that is 54 m high. The police at the scene of the accident note that the point of impact is 130 m from the base of the cliff. How fast was the car traveling when it went over the cliff?

3. A ball thrown horizontally at 22.2 m/s from the roof of a building lands 36 m from the base of the building. How tall is the building?

4. A boy kicked a can horizontally from a 6.5 m high rock with a speed of 4.0 m/s. How far from the base of the rock the can land?

5. A pilot flying a constant 215 km/h horizontally in a low-flying helicopter, wants to drop secret documents into his contact"s open car which is traveling 155 km/h in the same direction on a level highway 78.0 m below. At what angle (to the horizontal) should the car be in his sights when the packet is released?

6. A ski jumper travels down a slope and leaves the ski track moving in the horizontal direction with a speed of 25 m/s. The landing incline falls off with a slope of 33º.
a. How long is the ski jumper air borne?
b. Where does the ski jumper land on the incline?

7. Stones are thrown horizontally with the same velocity. One stone lands twice as far as the other stone. What is the ratio of the height of the taller building to the height of the shorter?

8. A fleck moving horizontally to the right at 2.5 m/s begins to accelerate downward at 0.75 m/s2 . Where is the fleck 4.0 s later?

B General Projectile Motion

1. In example 2, if Erica kicked the ball from the edge of the roof of a building which is 30.0 m high.
a. When does it strike the ground?
b. How far from the building does it land?

2 . A daredevil decides to jump a canyon of width 10 m. To do so, he drives a motorcycle up an incline sloped at an angle of 15 degrees. What minimum speed must he have in order to clear the canyon?

3. A ball is kicked from a point 38.9 m away from the goal. The crossbar is 3.05 m high. If the ball leaves the ground with a speed of 20.4 m/s at an angle of 52.2º to the horizontal
a. By how much does the ball clear or fall short of clearing the crossbar?
b. What is the vertical velocity of the ball at the time it reaches the crossbar?

4. A rocket is accelerating vertically upward at 30 m/s2 near Earth's surface. A bolt separates from the rocket. What is the acceleration of the bolt?

5. Water is leaving a hose at 6.8 m/s. If the target is 2 m away horizontally, What angle should the water have initially?

6. A 5.0 kg brick lands 10.1 m from the base of a building. If it was given an initial velocity of 8.6 m/s [61º above the horizontal], how tall is the building?

7. A spear is thrown upward from a cliff 48 m above the ground. Given an initial speed of 24 m/s at an angle of 30º to the horizontal,
a. how long is the spear in flight?
b. what is the magnitude and direction of the spear's velocity just before it hits the ground?

8. A projectile is shot from the edge of a cliff 125 m above ground level with an initial speed of 65.0 m/s at an angle of 37º above the horizontal. Determine the the magnitude and the direction of the velocity at the maximum height.

9. A projectile leaves a gun at the same instant that the target is dropped from rest. If the projectile is initially aimed straight at the target, will it hit the target?

10. A basketball is lobbed toward a hoop 3.05 m above the floor. If released 2 m above the floor 10 m from the basket and at a 45 degree angle, how fast must the basketball be thrown so that it goes through the hoop?

11. Dick is tossing chocolates up to Jane's window from 8.0 m below her window and 9.0 m from the base of the wall. If the chocolates are traveling horizontally through the open window, how fast are they going through her window?

12. A projectile has an initial velocity of 15.0 m/s at an angle of 30 degrees above the horizontal. What is the location of the projectile 2.0 seconds later?

13. If a ball is kicked with an initial velocity of 25 m/s at an angle of 60° above the ground, what is the "hang time"?

14. A water balloon hits a target 26 m away, at the same height as the release point. The horizontal component of the initial velocity was 5 m/s. What was the vertical component of the initial velocity? What was the launch angle?

15. A soccer ball leaves a cliff 20.2 m above the valley floor, at an angle of 10 degrees above the horizontal. The ball hits the valley floor 3.0 seconds later. What is the initial velocity of the ball? What maximum height above the cliff did the ball reach?

16. A flea stands 2.00 m from a dog's haunches .55m in height. Jumping at an angle of 32 degrees, what initial speed must the flea have to reach her new home?

17. A bullet hit a target 301.5m away. What maximum height above the muzzle did the bullet reach if it was shot at an angle of 25 degree to the ground?

18. A 3.00 kg parcel is dropped out of a window from a height of 176.4 m. Wind exerts an average 12.0 N force on the parcel away from the building. How long is the parcel in the air? Where does it land? What is its impact velocity?

19. A projectile is shot from the ground at an angle of 60 degrees with respect to the horizontal, and it lands on the ground 5 seconds later. Find:
a. the horizontal component of initial velocity
b. the vertical component of initial velocity
c. initial speed

20. An arrives 30m away horizontally and 5m above the point from which it was launched. It reaches this point 3 seconds after it was launched. Find:
a. the horizontal component of initial velocity
b. the vertical component of initial velocity
c. the vertical component of the impact velocity
d. the horizontal component of the impact velocity

21. Find the minimum initial speed of a champagne cork that travels a horizontal distance of 11 meters.

22. During practice, a soccer player kicks a ball, giving it a 32.5 m/s initial speed. It travels the maximum possible distance before landing down field.
(a) How much time does the ball spend in the air?
(b) How far did the ball travel?

23. A projectile was launched 64° above the horizontal, attaining a height of 10 m. What is the projectile's initial speed?

24. At what launch angle will the range of a projectile equal its maximum height?

25. A boy kicks a soccer ball directly at a wall 41.8 m away. The ball leaves the ground at 42.7 m/s with an angle of 33.0 degrees to the ground. What height will the ball strike the wall?

26. What is the relationship between the maximum height of the projectile, the projectile's range, and the launch angle?

27. A projectile is fired with an initial velocity of 120 m/s at an angle above the horizontal. If the projectile's initial horizontal speed is 55 m/s, then at what angle was it fired?

28. A boulder rolls 35 m down a hill, starting from rest and accelerating at 3.06 m/s2. The boulder then rolls off a 45 m high vertical cliff, launching at 19.0° below the horizontal. (a) How far from the cliff's base does the boulder land? (b) How much time does the boulder spend falling?